On anomalous diffusion in the Kraichnan model and correlated-in-time variants
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866909214270554112 |
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| author | Rowan, Keefer |
| author_facet | Rowan, Keefer |
| contents | We provide a concise PDE-based proof of anomalous diffusion in the Kraichan model -- a stochastic, white-in-time model of passive scalar turbulence. That is, we show an exponential rate of $L^2$ decay in expectation of a passive scalar advected by a certain white-in-time, correlated-in-space, divergence-free Gaussian field, uniform in the initial data and the diffusivity of the passive scalar. Additionally, we provide examples of correlated-in-time versions of the Kraichnan model which fail to exhibit anomalous diffusion despite their (formal) white-in-time limits exhibiting anomalous diffusion. As part of this analysis, we prove that anomalous diffusion of a scalar advected by some flow implies non-uniqueness of the ODE trajectories of that flow. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_12147 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On anomalous diffusion in the Kraichnan model and correlated-in-time variants Rowan, Keefer Mathematical Physics Analysis of PDEs Probability We provide a concise PDE-based proof of anomalous diffusion in the Kraichan model -- a stochastic, white-in-time model of passive scalar turbulence. That is, we show an exponential rate of $L^2$ decay in expectation of a passive scalar advected by a certain white-in-time, correlated-in-space, divergence-free Gaussian field, uniform in the initial data and the diffusivity of the passive scalar. Additionally, we provide examples of correlated-in-time versions of the Kraichnan model which fail to exhibit anomalous diffusion despite their (formal) white-in-time limits exhibiting anomalous diffusion. As part of this analysis, we prove that anomalous diffusion of a scalar advected by some flow implies non-uniqueness of the ODE trajectories of that flow. |
| title | On anomalous diffusion in the Kraichnan model and correlated-in-time variants |
| topic | Mathematical Physics Analysis of PDEs Probability |
| url | https://arxiv.org/abs/2311.12147 |